#### Glasnik matematički, Vol. 58 No. 1, 2023.

Izvorni znanstveni članak

https://doi.org/10.3336/gm.58.1.03

On the $$D(4)$$-pairs $$\{a, ka\}$$ with $$k\in \{2,3,6\}$$

Kouèssi Norbert Adédji orcid.org/0000-0001-8257-5729 ; Institut de Mathématiques et de Sciences Physiques, Université d'Abomey-Calavi, Bénin
Marija Bliznac Trebješanin orcid.org/0000-0003-0640-5407 ; Faculty of Science, University of Split, 21000 Split, Croatia
Alan Filipin ; Faculty of Civil Engineering, University of Zagreb, 10000 Zagreb, Croatia
Alain Togbé orcid.org/0000-0002-5882-936X ; Department of Mathematics and Statistics, Purdue University Northwest, 1401 S, U.S. 421, Westville IN 46391, USA

Puni tekst:

str. 35-57

preuzimanja: 186

###### Sažetak

Let $$a$$ and $$b=ka$$ be positive integers with $$k\in \{2, 3, 6\},$$ such that $$ab+4$$ is a perfect square. In this paper, we study the extensibility of the $$D(4)$$-pairs $$\{a, ka\}.$$ More precisely, we prove that by considering families of positive integers $$c$$ depending on $$a,$$ if $$\{a, b, c, d\}$$ is a set of positive integers which has the property that the product of any two of its elements increased by $$4$$ is a perfect square, then $$d$$ is given by
-17ex d=a+b+c+1/2(abc±√((ab+4)(ac+4)(bc+4))).
As a corollary, we prove that any $$D(4)$$-quadruple tht contains the pair $$\{a, ka\}$$ is regular.

304389

20.6.2023.

Posjeta: 433 *